The Dirichlet problem and Kakutani’s theorem

Abstract

[en] In this memoir we prove a weak version in $\mathbb{R}^2$ of Kakutani's theorem which gives a solution to the Dirichlet problem. The Dirichlet problem is a classical problem in partial differential equations with many applications in various fields. Given a bounded domain $D \subset$ $\mathbb{R}^d$ and a function $f$ continuous at $\partial D$, the Dirichlet problem consists in finding an harmonic function $u$ on $D$, which matches the values of $f$ on the boundary. It is known that for very general domains the solution exists and is unique. The solution given by Kakutani in 1944 is based in the use of probabilistic methods, specifically in the properties of Brownian motion, which will play an important role throughout this memoir.